Find Matrix Representation of Linear Transformation From $\R^2$ to $\R^2$

Ohio State University exam problems and solutions in mathematics

Problem 370

Let $T: \R^2 \to \R^2$ be a linear transformation such that
\[T\left(\, \begin{bmatrix}
1 \\
1
\end{bmatrix} \,\right)=\begin{bmatrix}
4 \\
1
\end{bmatrix}, T\left(\, \begin{bmatrix}
0 \\
1
\end{bmatrix} \,\right)=\begin{bmatrix}
3 \\
2
\end{bmatrix}.\] Then find the matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$ for every $\mathbf{x}\in \R^2$, and find the rank and nullity of $T$.

(The Ohio State University, Linear Algebra Exam Problem)
 
LoadingAdd to solve later

Solution.

The matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$ is given by
\[A=[T(\mathbf{e}_1), T(\mathbf{e}_2)],\] where $\mathbf{e}_1=\begin{bmatrix}
1 \\
0
\end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}
0 \\
1
\end{bmatrix}$ are standard basis of $\R^2$.
Since the vector $T(\mathbf{e}_2)$ is given, it remains to find $T(\mathbf{e}_1)$.
By inspection, we obtain the linear combination
\[\begin{bmatrix}
1 \\
0
\end{bmatrix}=\begin{bmatrix}
1 \\
1
\end{bmatrix}-\begin{bmatrix}
0 \\
1
\end{bmatrix}.\] Thus, we have
\begin{align*}
T(\mathbf{e}_1)&=T\left(\,\begin{bmatrix}
1 \\
1
\end{bmatrix}-\begin{bmatrix}
0 \\
1
\end{bmatrix} \,\right)\\
&=T\left(\, \begin{bmatrix}
1 \\
1
\end{bmatrix} \,\right)-T\left(\, \begin{bmatrix}
0 \\
1
\end{bmatrix} \,\right) && \text{by linearity of $T$}\\
&=\begin{bmatrix}
4 \\
1
\end{bmatrix}-\begin{bmatrix}
3 \\
2
\end{bmatrix}\\
&=\begin{bmatrix}
1 \\
-1
\end{bmatrix}.
\end{align*}
It follows that the matrix $A$ is given by
\[A=\begin{bmatrix}
1 & 3\\
-1& 2
\end{bmatrix}.\]

The rank and nullity of $T$ are the same as the rank and nullity of $A$.
We reduced the matrix $A$ by elementary row operations as follows:
\begin{align*}
A\xrightarrow{R_2+R_1} \begin{bmatrix}
1 & 3\\
0& 5
\end{bmatrix}
\xrightarrow{\frac{1}{5}R_2}
\begin{bmatrix}
1 & 3\\
0& 1
\end{bmatrix}
\xrightarrow{R_1-3R_2}
\begin{bmatrix}
1 & 0\\
0& 1
\end{bmatrix}.
\end{align*}
Hence the rank of $A$ is $2$ (because there are two non zero rows). The nullity of $A$ is determined by the rank nullity theorem
\[\text{rank of $A$} + \text{ nullity of $A$}=2 \text{ (the number of columns of $A$)}.\] Hence the nullity of $A$ is $0$.

In a nutshell, the rank of $T$ is $2$, and the nullity of $T$ is $0$.

Linear Algebra Midterm Exam 2 Problems and Solutions


LoadingAdd to solve later

Sponsored Links

More from my site

You may also like...

6 Responses

  1. 04/06/2017

    […] Problem 7 and its solution: Find matrix representation of linear transformation from $R^2$ to $R^2$ […]

  2. 04/07/2017

    […] Problem 7 and its solution: Find matrix representation of linear transformation from $R^2$ to $R^2$ […]

  3. 04/07/2017

    […] Problem 7 and its solution: Find matrix representation of linear transformation from $R^2$ to $R^2$ […]

  4. 04/07/2017

    […] Problem 7 and its solution: Find matrix representation of linear transformation from $R^2$ to $R^2$ […]

  5. 04/07/2017

    […] Problem 7 and its solution: Find matrix representation of linear transformation from $R^2$ to $R^2$ […]

  6. 04/07/2017

    […] Problem 7 and its solution: Find matrix representation of linear transformation from $R^2$ to $R^2$ […]

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

More in Linear Algebra
Ohio State University exam problems and solutions in mathematics
Rank and Nullity of Linear Transformation From $\R^3$ to $\R^2$

Let $T:\R^3 \to \R^2$ be a linear transformation such that \[ T(\mathbf{e}_1)=\begin{bmatrix} 1 \\ 0 \end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix} 0 \\ 1...

Close